Do black holes have a characteristic size?
Your lower bound has certainly been considered. Even Wikipedia does the calculation in a number of places, noting that a black hole will be in equilibrium with the $2.7\ \mathrm{K}$ CMB at about the mass of the Moon.
The thing about black holes, though, is that there are multiple channels for them to be produced in nature. The remnants of collapsed stars should have masses around $1\ M_\odot$. This number probably varies far less than the masses of stars themselves vary, but it is set by very complicated physics involving degenerate matter and stellar evolution as a whole.
On the other hand, supermassive black holes in the centers of galaxies have masses well over $10^9\ M_\odot$. So right there you're looking at at least $9$ orders of magnitude of observed variation. Add to that another $7$ or so on the lower end if you believe lunar-mass black holes exist, and you can see why there isn't a standard $1\ M_\mathrm{BH}$ unit.
It should be noted that most all black hole properties of interest scale perfectly in proportion to some power of the mass, usually just the first power. Thus one often normalizes quantities to the mass of the particular black hole in question.
Expanding on user10851's answer:
Black holes are thought to basically form originally in one of two ways. Primordial black holes would be created during the early Universe. I'm a bit rusty on this kind of physics, but my understanding is they could be of in principle any mass range. However, observational evidence limits them; I forget exactly how but I remember hearing numbers between 1 lunar mass and 100 solar masses. This is a little fringy, but possible.
Stellar-mass black holes form from the gravitational collapse of supermassive stars. They would have masses between about 5 to perhaps 100 solar masses. The masses are limited at the bottom by the size of progenitor star you need to get the collapse going (ending up with a BH and not a neutron star) and at the top by the maximum stellar mass and the amount of gas you shed during the supernova stage of the collapse. The error bars are pretty big here, largely because supernovas are poorly understood.
A BH of less than 5 solar masses would be a pretty big discovery, by the way - not a Nobel Prize or anything, but for sure a Nature paper - since the formation from such lower-mass stars is though to lead to a neutron star, not a black hole. Thus, there is expected to be a compact-object "mass gap" between 2 (the maximum NS mass) and 5 solar masses.
For essentially thermodynamic reasons, in dense stellar environments (globular clusters and galactic cores) heavy objects like black holes are supposed to sink into the centre of the gravitational potential, where they tend to merge with one another, forming black holes with larger masses. Over millennia as galaxies and clusters collide, these large central objects repeatedly merge, forming eventually supermassive black holes of $10^7$ - $10^{10}$ solar masses.
There are, perhaps, also intermediate mass black holes of between 100-1 million solar masses, which would form by similar device as their supermassive counterparts, but continued for fewer mergers.